Surveying Calculating Irregular Parabolic Area THE TRAPEZOIDAL RULE

Surveying Calculating Irregular Parabolic Area

THE TRAPEZOIDAL RULE While applying the trapezoidal rule, boundaries between the ends of ordinates are assumed to be straight. Thus the areas enclosed between the base line and the irregular boundary line are considered as trapezoids. Let O 1, O 2, …. . On=ordinate at equal intervals, and d= common distance between two ordinates

THE TRAPEZOIDAL RULE Thus the trapezoidal rule may be stated as follows: • To the sum of the first and last ordinate, twice the sum of intermediate ordinates is added. • This total sum is multiplied by the common distance. • Half of this product is the required area.

SIMPSON’S RULE In this rule, the boundaries between the ends of ordinates are assumed to form an arc of parabola. Hence simpson’s rule is some times called as parabolic rule. Refer to figure:

SIMPSON’S RULE Let O 1, O 2, O 3= three consecutive ordinates d= common distance between the ordinates area AFe. DC= area of trapezium AFDC+ area of segment Fe. DEF

SIMPSON’S RULE Total area = d/3[O 1+On+4(O 2+O 4+……) + 2(O 3+O 5)] Thus the rule may be stated as the follows • To the sum of the first and the last ordinate, four times the sum of even ordinates and twice the sum of the remaining odd ordinates are added. • This total sum is multiplied by the common distance. • One third of this product is the required area.

Comparison of Trapezoidal and Simpsons Rule Trapezoidal rule Simpson’s rule • The boundary between the ordinates is considered to be straight • There is no limitation. • It can be applied for any number of ordinates • It gives an approximate result • The boundary between the ordinates is considered to be an arc of a parabola • To apply this rule, the number of ordinates must be odd • It gives a more accurate result.

Example: Trapezoidal Rule and Simpsons Rule The following offsets were taken from a chain line to an irregular boundary line at an interval of 10 m: 0, 2. 50, 3. 50, 5. 00, 4. 60, 3. 20, 0 m Compute the area between the chain line, the irregular boundary line and the end of offsets by: a) the trapezoidal rule b) Simpson’s rule

Solution: Trapezoidal Rule and Simpsons Rule By trapezoidal rule: Here d=10 m Required area=10/2{0+0+2(2. 50+3. 50+5. 00+4. 60+3. 20+)} = 5*37. 60=188 m 2

Solution: Trapezoidal Rule and Simpsons Rule By Simpson’s rule: d=10 m Required area=10/3{0+0+4(2. 50+5. 00+3. 20)+2(3. 50+4. 60)} = 10/3{ 42. 80+16. 20}=10/3*59. 00 10/3*59= 196. 66 m 2

Example 2: Trapezoidal and Simpsons Rule The following offsets were taken at 15 m intervals from a survey line to an irregular boundary line 3. 50, 4. 30, 6. 75, 5. 25, 7. 50, 8. 80, 7. 90, 6. 40, 4. 40, 3. 25 m Calculate the area enclosed between the survey line, the irregular boundary line, and the offsets, by: Trapezoidal and Simpsons Rule.

Solutions: Trapezoidal Rule The trapezoidal rule: Required area=15/2{3. 50+3. 25+2(4. 30+6. 75+5. 25+7. 50+ 8. 80+7. 90+6. 40+4. 40)} = 15/2{6. 75+102. 60} = 820. 125 m 2

Solutions: Simpsons Rule • If this rule is to be applied, the number of ordinates must be odd. But here the number of ordinate is even(ten). • So, Simpson's rule is applied from O 1 to O 9 and the area between O 9 and O 10 is found out by the trapezoidal rule.

Solutions: Simpsons Rule A 1=15/3{3. 50+4. 40+4(4. 30+5. 25+8. 80+6. 40) +2(6. 75+7. 50+7. 90) } = 15/3(7. 90+99. 00+44. 30) = 756. 00 m 2 A 2= 15/2(4. 40+3. 25) = 57. 38 m 2 Total area= A 1+ A 2 =756. 00+57. 38 = 813. 38 m 2

Class Exercise 1: I. Calculate the total area of a field with a base of 50 m and ordinates 5, 6, 5, 7, 8, 9, 6. 5, 7. 5, 8, 8. 5 and 9 m respectively. Use Simpson’s rule for your calculations. If the selling price of the field is $150 per square kilometer, what is the total cost of that field?

Class Exercise 2: Calculate the total area of a field with a base of 60 m and ordinates 5, 6, 5, 7, 8, 9, 6. 5, 5, 7. 5, 8, 8. 5 and 9 m respectively. Calculate the area of the field using Trapezoidal and Simpsons Rule

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